Abstract
In this lesson plan, students examine the patterns that emerge from repetitive behavior. First, students will consider fractals, exploring the concept of recursion. Then, students will explore an agent model, where an overall pattern emerges from repetitive individual behaviors.
Objectives
Upon completion of this lesson, students will:
- understand the concept of recursion
- understand how patterns emerge from repetition
- understand the basic principles of computer modeling
Standards Addressed
Grade 3
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Geometry
- The student demonstrates an understanding of geometric relationships.
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates understanding of position and direction.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 4
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Geometry
- The student demonstrates an understanding of geometric relationships.
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates understanding of position and direction.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Grade 5
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Geometry
- The student demonstrates an understanding of geometric relationships.
- The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
- The student demonstrates understanding of position and direction.
- The student demonstrates a conceptual understanding of geometric drawings or constructions.
Fourth Grade
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Operations and Algebraic Thinking
- Generate and analyze patterns.
Fifth Grade
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Operations and Algebraic Thinking
- Analyze patterns and relationships.
3rd Grade
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Geometry
- The student will demonstrate through the mathematical processes an understanding of the connection between the identification of basic attributes and the classification of two-dimensional shapes.
5th Grade
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Geometry
- The student will demonstrate through the mathematical processes an understanding of congruency, spatial relationships, and relationships among the properties of quadrilaterals.
4th Grade
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Geometry
- Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and movement within the first quadrant of a coordinate system.
4th Grade
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Geometry
- 4.17.c The student will investigate congruence of plane figures after geometric transformations such as reflection (flip), translation (slide) and rotation (turn), using mirrors, paper folding, and tracing.
5th Grade
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Geometry
- 5.15a The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize, identify, describe, and analyze their properties in order to develop definitions of these figures
- 5.15e The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize the images of figures resulting from geometric transformations such as translation (slide), reflection (flip), or rotation (turn).
Student Prerequisites
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Technological:
Students must be able to:
- perform basic mouse manipulations such as point, click, and drag
- use a browser for experimenting with the activities
Teacher Preparation
Teachers will need:
- access to a java-enabled browser
- pencil for each student
- a copy of the worksheet for each student
Key Terms
fractal
Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration
pattern
Characteristic(s) observed in one item that may be repeated in similar or identical manners in other items
recursion
Given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information
self-similarity
Two or more objects having the same characteristics. In fractals, the shapes of lines at different iterations look like smaller versions of the earlier shapes
Lesson Outline
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Focus and Review
Introduce patterns by asking students to give examples of patterns they've seen recently.
- If students are having trouble naming things, you might point out clothing with patterns, for example: "Will's shirt has a plaid pattern on it".
- Ask students how they know these things are patterns.
- Lead the students in a discussion about patterns.
Introduce the concept of recursion by leading the students in a discussion.
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Ask students if they can think of recursive patterns.
- For example: 1, 2, 4, 8, 16; where each step doubles to create the next step
- Ask students how these patterns are different from the patterns discussed earlier.
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Objectives
Let the students know what it is they will be doing and learning today. Say something like this:
- Today, class, we are going to learn about recursion, modeling, and patterns.
- We are going to use the computers to learn about this, but please do not turn your computers on until I ask you to. I want to show you a little about these activities first.
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Teacher Input
Draw an equilateral triangle on the board. Tell the students that you're going to make a recursive pattern with this triangle by replacing each line segment with the design shown below.
Ask students what they expect the pattern to look like. Now draw the new shape on the board and
ask the students to imagine what the shape would look like if you repeated the rule again.
Have student volunteers come up to draw each step. When the process starts to take too long for each step (after the third or fourth iterate), pull up the Koch's Snowflake applet. Click through the steps to watch the snowflake start to form. Discuss the results as a class:
- Was this the pattern you were expecting?
- How is this pattern different from the patterns we talked about at the beginning of our discussion?
- Do you see the recursion in this pattern?
Take a moment to ask the students how they think the computer helped them in creating this pattern. Help them see that when doing lots of repetition, the computer can do the repetitive calculations over and over so that the thinking human beings can analyze the results.
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Guided Practice
Lead a discussion on agent modeling. Then introduce the Rabbits and Wolves applet. Discuss how to use the applet, specifically showing students the population graph. Do not discuss the individual agents or their behaviors yet.
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Independent Practice
Have the students work in pairs to run the simulation and answer the questions on the worksheet.
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Closure
As a class, discuss the findings of the activity. Ask the students the following guided questions:
- What patterns did you notice?
- Would you have noticed those if you had only looked at an individual agent?
- Would you have noticed those if you had only looked at the populations at one given moment?
Alternate Outline
If only one computer is available for the classroom, this lesson can be rearranged in the following way:
- Project the Rabbits and Wolves applet for the whole class to see.
- Have students work in pairs to complete the worksheet as you run the simulation for the whole class to observe.
- Be sure to have the students observe multiple trials of the simulation in order to draw accurate conclusions.
Suggested Follow-Up
Students can explore more about modeling in the Fire: Modeling Probability lesson plan.